Adaptive pinning security synchronization for multi-layer networks under asynchronous link attacks

In response to the complex security threats faced by multi-layer networks (MLNs), this paper investigates the security synchronization problems under asynchronous link attacks, where each layer consists of identical nonlinear node dynamics and inter-layer couplings are bidirectional. Unlike existing works that assume isolated or synchronous attack patterns, we construct a unified link attack model in which intra-layer links and inter-layer links are compromised independently by switching signals with unknown bounds, asynchronous switching instants and bounded attack durations. To compensate for the resulting coupling degradation, we develop an adaptive pinning control strategy equipped with an online state-driven update law for each pinned node. Using Lyapunov stability theory, we prove that the proposed strategy can achieve intra-layer security synchronization (ALSS) and inter-layer security synchronization (RLSS), respectively. Sufficient synchronization conditions are derived as scalar inequalities that explicitly depend on the attack duration. Finally, two numerical simulations verify the effectiveness of the proposed method.